Journal of Applied MathematicsVolume 2011 (2011), Article ID 378389, 19 pagesdoi:10.1155/2011/378389

Research Article

Periodic Solutions for Autonomous -Laplacian System with Impulsive Effects

Xiaoxia Yang and Haibo Chen

School of Mathematical Sciences and Computing Technology, Central South University, Changsha, Hunan 410083, China

Received 17 July 2011; Accepted 31 August 2011

Academic Editor: Yongkun Li

Copyright © 2011 Xiaoxia Yang and Haibo Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

By using the variational method, some existence theorems are obtained for periodic solutions of autonomous -Laplacian system with impulsive effects.

1. Introduction

Let , , .

In this paper, we consider the following system: where , , ,  , and are the instants where the impulses occur and , , and are continuously differentiable and satisfies the following assumption. is continuously differentiable in , and there exist such that

for all .

When , , , and , system (1.1) reduces to the following autonomous second-order Hamiltonian system: There have been lots of results about the existence of periodic solutions for system (1.4) and nonautonomous second order Hamiltonian system (e.g., see [1–21]). Many solvability conditions have been given, for instance, coercive condition, subquadratic condition, superquadratic condition, convex condition, and so on.

When , , , and , system (1.1) reduces to the following autonomous second-order Hamiltonian system with impulsive effects: Recently, many authors studied the existence of periodic solutions for impulsive differential equations by using variational methods, and lots of interesting results have been obtained. For example, see [22–28]. Especially, nonautonomous second-order Hamiltonian system with impulsive effects is considered in [25, 26] by using the least action principle and the saddle point theorem.

When and , system (1.1) reduces to the following system: In [29, 30], Paşca and Tang obtained some existence results for system (1.7) by using the least action principle and saddle point theorem. Motivated by [17, 22–30], in this paper, we are concerned with system (1.1) and also use the least action principle and saddle point theorem to study the existence of periodic solution. Our results still improve those in [17] even if system (1.1) reduces to system (1.4).

A function is called to be -quasiconcave if for some and .

Next, we state our main results.

Theorem 1.1. Let and be such that and . Suppose satisfies assumption (A) and the following conditions: there exist such that where , there exists such that Then, system (1.1) has at least one solution in , where is absolutely continuous, and. Furthermore, if , and the following condition holds: there exist , and such that then system (1.7) has at least two nonzero solutions in .

When , , by Theorem 1.1, it is easy to get the following corollary.

Corollary 1.2. Suppose satisfies the following conditions: is continuously differentiable in and there exists such that for all . there exists such that ; there exists such that Then, system (1.6) has at least one solution in . Furthermore, if and the following condition holds: there exist and such that then system (1.4) has at least two nonzero solutions in .

For the Sobolev space , one has the following sharp estimates (see in [3, Proposition  1.2]): By the above two inequalities, we can obtain the following better results than by Corollary 1.2.

Theorem 1.3. Suppose satisfies assumption (A)′, (F2)′, (I1)′ and there exists such that (1.15) holds.Then, system (1.6) has at least one solution in . Furthermore, if and the following condition holds: there exist and such that then system (1.4) has at least two nonzero solutions in .

Moreover, for system (1.6), we have the following additional result.

Theorem 1.4. Suppose satisfies assumption (A)′, (F1)′ ′ and the following conditions: is -quasiconcave on , as , , there exist such that there exist , , such that Then, system (1.6) has at least one solution in .

Remark 1.5. In [17], Yang considered the second-order Hamiltonian system with no impulsive effects, that is, system (1.4). When , our Theorems 1.3 and 1.4 still improve those results in [17]. To be precise, the restriction of is relaxed, and some unnecessary conditions in [17] are deleted. In [17], the restriction of is , which is not right. In fact, from his proof, it is easy to see that it should be . Obviously, our restriction is better. Moreover, in our Theorem 1.4, we delete such conditions (of in [17, Theorem  1]): , and there exist positive constants such that Finally, it is remarkable that Theorems 1.3 and 1.4 are also different from those results in [1–16]. We can find an example satisfying our Theorem 1.3 but not satisfying the results in [1–21]. For example, let where . We can also find an example satisfying our Theorem 1.4 but not satisfying the results in [1–21]. For example, let where .

2. Variational Structure and Some Preliminaries

The norm in is defined by Set Let Obviously, is a reflexive Banach space. It is easy to know that is a subset of and . In this paper, we will use the space defined by with the norm . It is clear that is a reflexive Banach space. Let . Then, .

Lemma 2.1 (see [31] or [32]). Each and each can be written as and with Then, where

Note that if , then is absolutely continuous. However, we cannot guarantee that is also continuous. Hence, it is possible that , which results in impulsive effects.

Following the idea in [22], one takes and multiplies the two sides of

by and integrate from 0 to , one obtains Note that is continuous. So, . Combining , one has Combining with (2.10), one has Similarly, one can get for all . Considering the above equalities, one introduces the following concept of the weak solution for system (1.1).

Definition 2.2. We say that a function is a weak solution of system (1.1) if holds for any . Define the functional by where , By assumption (A) and [33], we know that . The continuity of and implies that . So, , and for all , we have Definition 2.2 shows that the critical points of correspond to the weak solutions of system (1.1).We will use the following lemma to seek the critical point of .

Lemma 2.3 (see [3, Theorem  1.1]). If is weakly lower semicontinuous on a reflexive Banach space and has a bounded minimizing sequence, then has a minimum on .

Lemma 2.4 (see [34]). Let be a function on with , satisfying (PS) condition, and assume that for some , Assume also that is bounded below and , then has at least two nonzero critical points.

Lemma 2.5 (see [35, Theorem  4.6]). Let , where is a real Banach space and and is finite dimensional. Suppose that satisfies (PS)-condition and there is a constant and a bounded neighborhood of 0 in such that , there is a constant such that . Then, possesses a critical value . Moreover, can be characterized as where,

3. Proof of Theorems

Lemma 3.1. Under assumption (A), is weakly lower semicontinuous on .

Proof. Let Since then is convex. Moreover, by [33], we know that is continuous, and so, it is lower semicontinuous. Thus, it follows from [3, Theorem  1.2] that is weakly lower continuous. By assumption (A), it is easy to verify that is weakly continuous. We omit the details. Let Next, we show that and are weakly continuous on and , respectively. In fact, if then by in [3, Proposition  1.2], we know that So, there exists such that and , for all . Thus, we have Hence, is weakly continuous on . Similarly, we can prove that is also weakly continuous on . Thus, we complete the proof.

Proof of Theorem 1.1. It follows from and (2.7) that Hence, by , (3.7), and (3.8), we have Note that for , and for , So, and (3.9) imply that Thus, by Lemma 2.3, we know that has at least one critical point which minimizes on . Furthermore, if and , then system (1.1) reduces to (1.7). When also holds, we will use Lemma 2.4 to obtain more critical points of . Let ,    and . By (3.9), we know that as . So, satisfies (PS) condition and is bounded below. Take , where is a positive constant such that and for all . It follows from and Lemma 2.1 that Since and , (3.13) implies that for all with . By , it is easy to obtain that , for all with .If , then from above, we have for all with . Hence, all with are minimal points of , which implies that has infinitely many critical points. If , then by Lemma 2.4, has at least two nonzero critical points. Hence, system (1.7) has at least two nontrivial solutions in . We complete our proof.

Proof of Theorem 1.3. We only need to use (1.18) and (1.19) to replace (2.6) and (2.7) in the proof Theorem 1.1 with , and . It is easy. So, we omit it.

Lemma 3.2. Under the assumptions of Theorem 1.4, the functional defined by satisfies (PS) condition.

Proof. Suppose that is a (PS) sequence for ; that is, there exists such that Hence, for large enough, we have . It follows from , , and (1.18) that for large enough. By (1.18), we have and (3.16), (3.17), and imply that there exists such that It follows from , (3.15), (I3), (1.18), and (3.18) that for all and (3.19) and (F5) imply that is bounded. Combining (3.18), we know that is a bounded sequence. Similar to the argument in [25], it is easy to obtain that satisfies (PS) condition.Proof of Theorem 1.4. From and , it is easy to see that for , For all , by (1.18), and , we have Note that for all , is equivalent to . Then, , and (3.21) imply that It follows from (3.20) and (3.22) that satisfies and in Lemma 2.5. Combining with Lemma 3.2, Lemma 2.5 shows that has at least one critical point. Thus, we complete the proof.

4. Examples

Example 4.1. Let , , , , and . Consider the following system: where , , , , , . It is easy to verify that all conditions of Theorem 1.1 hold so that system (4.1) has at least one weak solution. Moreover, if , , and , , then system (4.1) has at least two nonzero solutions.

Example 4.2. Let , . Consider the following autonomous second-order Hamiltonian system with impulsive effects: where ,, . It is easy to verify that all conditions of Theorem 1.3 hold so that system (4.2) has at least one weak solution. Moreover, if and , , then system (4.2) has at least two nonzero solutions.

Example 4.3. Let , . Consider the following autonomous second-order Hamiltonian system with impulsive effects: where , . It is easy to verify that all conditions of Theorem 1.4 hold so that system (4.3) has at least one weak solution.

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